Global Geometry of planar vector fields

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This proposal is on planar polynomial vector fields.  There are several hard problems on these systems  still unsolved after more than a century: two problems posed by Poincaré and one posed by Hilbert, the second part of his 16th problem. The proposed research is expected to throw light on these open problems and will also have an impact on applications.  The following points indicate the main research lines of this project.  ***I) Global studies of families of polynomial vector fields.  The class of quadratic differential systems is 5-dimensional modulo the action of the group of affine transformations and time rescaling.  So far only sub-families of dimension at most three have been studied from a truly global viewpoint, among them the 3-dimensional family of Lotka-Volterra systems, very important for applications, studied by the applicant together with N. Vulpe. Some other important sub-families will be studied, among them the 4-dimensional family of quadratic vector fields with a first order weak focus, important  for Hilbert's 16th problem. Some cubic families will also be studied.***II) The applicant will continue her study with her collaborators on the classification of singularities of the whole quadratic class. Together with Artés, Llibre and Vulpe she already published 4 papers on this topic, another paper has been accepted for publication and another one was submitted in July 2014.  There remain 6 cases to be studied and form part of this research proposal: the case with four distinct real finite singularities, with 4 distinct complex singularities, with two real and two complex singularities, with 3  real singularities, one of them double, with two complex simple and one real double singularities and with a single real singularity of multiplicity four. Over 1000 global configurations of singularities for the quadratic class are expected. This rather huge classification theorem will have applications. Indeed, in the study of models in  applied mathematics involving quadratic systems, our algorithm could be used for effective computation of the global configurations of the singularities of the systems involved, information  available at the touch of a button.***III) Another line of research is on problems of integrability of polynomial vector fields in terms of invariant algebraic curves. Poincaré's problem on algebraic integrability and problems resulting from the work of Darboux on integrability in terms of algebraic invariant curves are open.  The Theorem of Darboux (or of Jouanolou) gives only sufficient conditions for Darboux (respectively algebraic) integrability and the number of curves involved is not optimal. The applicant wants to detect necessary conditions for algebraic or Darboux integrability, involving the degrees and multiplicities of the curves, conditions on the their relative position such as  intersection multiplicities of the curves and conditions on their singularities.   **